Runge-Kutta Theory for Volterra and Abel Integral ...€¦ · VOLTERRA AND ABEL INTEGRAL EQUATIONS...

mathematics of computationvolume 4l number 163july 1983. pages k7-i02

Runge-Kutta Theory for Volterra and Abel

Integral Equations of the Second Kind*

By Ch. Lubich

Abstract. The present paper develops the local theory of general Runge-Kutta methods for a

broad class of weakly singular and regular Volterra integral equations of the second kind.

Further, the smoothness properties of the exact solutions of such equations are investigated.

1. Introduction. We consider the Volterra integral equation of the second kind

(1) y(x) =f(x) + (\x- s)"K(x,s,y(s))ds, x G / := [0, x], a > -1.A)

The function f: I — R" is assumed to be (at least) continuous, the kernel K: S X R"

-» R" with S = {(x, s) 10 < s < x < x} is to be sufficiently differentiable.

For -1 < a < 0 the integral equation (1) is weakly singular and sometimes called

an Abel integral equation of the second kind. The special case a = - ? (Abel

equation in the proper sense) often arises in physical problems (see the references in

[13] or [7]). Positive values of a are encountered in various biological models [2] and

in statistics [14].

There exist general local existence and uniqueness theorems, and we suppose that

the existence interval of y(x) is the whole of /. In Section 2 of this paper, we shall

give smoothness and analyticity properties of the solution.

For many proofs it will be convenient to assume that the kernel K(x, s, y) is

independent of s, i.e. K(x, s, y) = A^(x, y). This is no restriction of generality, since

otherwise we may take (x, y(x)) as the solution of the integral equation

(y(x)) = (f{x)) +Ox-^(K(x,(s°,y(s))))ds-

The simple relation

(2) f\h- s)"sßds = B(a + l,ß+ \)-hx+a+ß (a,ß>-l),

where B denotes the Beta-function [1], will often be used in this paper. As an almost

immediate consequence, product quadrature rules are of the form

m

(\h-s)ag(s)ds~hx+«ïu,g(c,h),J0 i=\

Received August 5, 1982; revised November 5, 1982.

1980 Mathematics Subject Classification. Primary 65R20.

* This work has been supported by the Deutsche Forschungsgemeinschaft.

©1983 American Mathematical Society

0025-5718/83/0000-1369/S04.75

87

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

88 CH LUBICH

where the u¡, c, do not depend on h. This suggests considering the following type of

Runge-Kutta methods for the numerical solution of (1):

m

(3) r/-) = Fn(xn + c,h) +hx+«2 a,JK(x„ + d:]h, x„ + Cjh, F/">)7=1

( z = 1,..., m ),

m

Ä+i = Fn(xn + h)+h]+°2 b,K(xn + e,h, xn + Cjh, F,<">),i=i

where x„ = nh and Fn(x) denotes an approximation to the lag term

Fn(x) = f(x) + [X"(x - s)"K(x, s, y(s)) ds.

The method is explicit if a¡j — 0 for 1 < i <j < m. For a = -§ such methods have

first been used by Oules [11]. For a = 0 Brunner, Hairer and Norsett [3] have

characterized the order of the local error of (3) in terms of the coefficients of the

method. Their paper has in many ways been a model for the present work (far

beyond the choice of the title). However, in the case of noninteger a an inherent lack

of smoothness does not permit a direct extension of the results and techniques of [3].

So a basically different approach to the order conditions is given in Section 3.

Finally, Section 4 contains a variety of examples of explicit and implicit Runge-Kutta

methods (3).

2. Smoothness Properties of the Solution. In order to construct numerical methods

for the approximate solution of integral equations (1), knowledge of the smoothness

properties of the exact solution is indispensable.

The following theorem states that the solution y(x) of (1) is smooth in any closed

interval bounded away from 0. It is a straightforward extension of a result in Miller

and Feldstein [9], and we state it without proof.

Theorem 1. Consider the integral equation (1). Assume that f(x) is continuous in

[0, x] and real analytic in (0, x), and let the kernel K(x, s, y) be real analytic in

S X R". Then the solution y(x) of (I) is real analytic in the open interval (0, x).

However, in general, v-(x) will not be analytic at x = 0. Apparently a complete

answer to the behavior at 0 is as yet unknown in the literature (see, e.g., the recent

papers [4], [5]). For example, Picard iteration shows that the integral equation

y(x) = 1 + [X(x-s)']/3y{s)ds

has the solution y(x) = 1 + |x2/3 + 0(x4/3) as x -> 0. The structure of these

singularities is well understood for the special case a = - {- (see Miller and Feldstein

[9], de Hoog and Weiss [7]). The following result characterizes the behavior of the

solution near 0 for arbitrary a > -1.

Theorem 2. Consider the integral equation (1). Suppose that f(x) — F(x, xx+a),

and assume that both F(zx, z2) and the kernel K are real analytic at the origin. Then

there is a function Y(zx, z2), real analytic at (0,0), such that y(x) = Y(x, x1+a).


VOLTERRA AND ABEL INTEGRAL EQUATIONS 8 9

Remark. The smoothness of y(x) at 0 is not improved if f(x) itself is real analytic.

On the contrary, it is easily seen that f(x) and y(x) cannot be smooth at 0

simultaneously (excluding, of course, the trivial case where a is an integer).

Proof. Without loss of generality we may assume K(x, s, y) = K(x, y) and

/(0) = 0. We first give the proof for the one-dimensional case. Let K(x, y) =

2Zk^QKk(x)yk. We take an arbitrary analytic function A(zx, z2) = 1nanzn (where

n = (/j,, n2) ranges over Nq\{(0,0)}, and z = (zx, z2)) and insert A(x, x1+Q) for

y(x) into the integral of (1):

f (x - s)aK(x, A(s, sx+a)) ds = f(x - s)a2Kk(x)Í2ans"Ísx+°)"Ak ds•>o Jo k \ „ I

= lKk(x)ZQkn(A)f(x-s)as"isx+")">ds,k n y0

where Qkn(A) is a polynomial in a^, axo, aox,...,ani„ having only nonnegative

coefficients. (The sums and integrals can be interchanged because of uniform

convergence.)

We now use formula (2) and write

I(n) =5(1 +0,1+1!, +«2(1 +a)) = (\l -t)at"itx+a)"2dt,Jo

so that the expression above reduces to

(4) r(x-syK(x,A(s,sx+a))ds = xi+"2l(n)?iKk(x)Qkn(A)x'"(xx+ay\J° n k

This and formula (1) indicate how we have to choose Y(zx, z2): We define Y as the

formal power series in z = (zx, z2) given by

(5) Y(zx,z2) = F(zx, z2) + z2^l(n)^Kk(zx)Qkn(Y)z".n k

The factor z2 at the right-hand side of (5) allows the recursive computation of the

coefficientsy„ of Y(z) = 2„ y„z".

As a next step we proceed to demonstrate that the formal solution Y(z) defined in

(5) actually represents a convergent power series and hence a (real) analytic function

in a neighborhood of (0,0): Let F and K denote convergent majorants of F and K,

respectively. Define the formal power series Y by

(6) Y(zx,z2) = F(zx, z2) + z2/(0)2 lKk(zx)Qkn(Y)z".n k

Observing |/(«) |< 7(0) for all n and the nonnegativity of the coefficients of the

polynomials Qkn, an easy induction argument shows that Y is a majorant of Y.

Moreover, we may rewrite (6) as

Y(z) = F(z) + z2l(0)K(zx,Y(z)),

and the analytic version of the implicit function theorem implies that Y(z), and

hence also Y(z), are analytic in a neighborhood of (0,0).

So we can finally use (4) and (5) to conclude that y(x) = F(x, xx+a) is indeed a

solution (and so, by uniqueness, the solution) of the integral equation (1) near 0.


90 CH LUBICH

In the higher-dimensional case the Kk(x) are symmetric /(-linear forms, and

expressions like Kk(x)yk have to be interpreted as Kk(x)(y.y). Then the above

proof carries immediately over to the general case. D

Corollary 3. If the function F with f(x) = F(x, x1 +Q) and the kernel K are only

assumed to be sufficiently differentiable, then the solution y( x ) of ( 1 ) has an asymptotic-

expansion in mixed powers of x and xx+a as x — 0.

Proof. We construct a truncated power series YN(zx, z2) as far as possible (say, of

degree TV) according to (5) and put yN(x) = YN(x, xx+a). Then (4) shows that the

defect

8(x) = yN(x) - f(x) - f\x - s)"K(x, yN(s)) dsJo

is of magnitude 0(xN) + 0((xx +af) as x - 0.

We may interpret the integral equation (1) as a nonlinear operator equation in

C[0, x] (equipped with the supremum norm):

y = f + T(y) and correspondingly yN—f+8-¥ T(yN).

The estimate

IITO) - r(z)|| < \\y - z\\l(\x - s)"ds,Jo

where L denotes a Lipschitz constant of the kernel K, shows that the Lipschitz

constant of Tcan be made smaller than one if x is chosen small enough. Subtracting

the two equations, we obtain that the error y(x) - yN(x) is of the same magnitude

as the defect. D

3. Order Conditions. The first part of this section is devoted to the study of the

local error of Runge-Kutta methods (3) in an interval bounded away from 0.

Without loss of generality (also with respect to (3)) we may assume that the kernel

A" in (1) is independent of .v. We fix x() in the open interval (0, x) and rewrite (1) as

(7) y(x) = F(x) + f\x-s)"K(x, y(.s))ds forxG[x„,x],

where F(x) =/(x) + Jt)"(x ~ s)"K(x, y(s))ds. Note that by Theorem 1 the solu-

tion y(x) is smooth at x0. Applying one step of the Runge-Kutta method (3) to the

integral equation (7) we obtain

m

(8) Yj = F(x0 + cih) + h>+°2aijK(x0 + djjh,Yj) (/= 1.m),7=1

m

yx = F(x0 + h) + hx+a 2 bjK(x0 + e,h, Y,).

The following two definitions will allow us to state in Theorem 6 purely algebraic

conditions on the coefficients of the Runge-Kutta method which imply that the local

errors, — y(x0 + h) is of a prescribed order.

As in [3], the following set of trees will play a decisive role.

Definition A. Let TV (Volterra-trees) denote the set of all trees which may or may

not have an index x attached to any of their final nodes.


VOLTERRA AND ABEL INTEGRAL EQUATIONS 91

For a tree t G TV we introduce

fin(z) = number of final nodes of t,

int(z) = number of interior nodes of t.

(The root is counted as an interior node.)

As in [3], we use for t,,...,t G TV the notation [t*, t', r,.tq] to designate a

new t G TV which is illustrated in Figures 1 and 2.

Figure 1

/ 7 Vt = [ ] [t] [tJ

V[t ] Itx,t]

.. <

not in TV

Figure 2

In Figure 3 we have marked the final nodes. Here we have fin(z) = 7, int(z) = 5.

Figure 3

Definition 5. Let J(l) = /0'(1 - s)as' ds for / > 0. We define functions y„ <p: TV -~>

R (z = \,...,m) recursively by

<p,(hM)= 2«zXcW(')c?+,+,+07=1

m

9(hk,r'})=lbjekc'j-J(l)i=i

(k,l>0),

and

<p,(0= 2 V.*c>>(/i)---«p>(/J>7=1

<p(0= 2 v^9,(í,)---<í>,(^),i=i

fOT/ = K*,T',í„...,/fl(/,^T,T„9>l).


92 CH. LUBICH

In the sequel we shall assume that <p,(r) = 0, i.e.

m

(9) 2 a,j = J(0)cx +a for i = 1,...,m.

7 = 1

(In the special case a = 0 this is the familiar condition 2ya,7 = c¡.) We are now in a

position to state the main result of this section.

Theorem 6. Consider an integral equation (1) whose kernel K and solution y are

sufficiently smooth at x0 (cf. Theorem 1). Then the condition

(10) <p(0 = 0 for all Z G TVwithim(t) + (1 + a)int(z) 0 which depends on the exponent a in (1).

Proof. Let K(h, s) = K(x0 + h, y(x0 + s)), and define the function

g(h) = ~-J\h - s)aK(h, s) ds = -~-f\h - s)a 2 ^rjdkdiK(0,0)hkS'dsh Jo h Jo k.i

= 2^9R'*(0,0)/z*+',k.i *•'•

which is seen to be smooth at h = 0.

As in the proof of Theorem 2, the basic idea is now to regard the functions

occurring in (8) as functions of two independent variables h, k. At the end of the

proof we shall then insert hx+a for k. We write formula (8) (with F inserted from (7))

as

17!

(11) y(h, K)=y(x0 + Cjh) - cx+aKg(c,h) + k 2 atJK{x0 + d:jh, y(h, k)),

7=1

m

yx(h, K)=y(x0 + h)- Kg(h) + k 2 b,K(x0 + e,h, Y,(h, k)).i=i

We have

(12) Y,= Yj(h,hx+"), yx=yx(h,hx+°),

(13) Yj(h,Q)=y(x0 + c,h), yx(h,0)=y(x0 + h),

and also

m

ZJ,(h,0) = -cx+"g(Cjh) + 2 atJK{xz + dtjh, y(xQ + Cjh)).7=1

This expression can be expanded into a Taylor series in h. This yields

U,(h,0)= 2 <P,([t;,t'])([t/,t'])^+',k,l>0

where <p, is given by Definition 5 and the $'s are expressions which only contain

derivatives of y and K at x0, but no longer depend on the coefficients of the

Runge-Kutta method.



Turning our attention to the higher derivatives of Yj in (11) with respect to k, we

obtain

m

KY,(h,0) = r 2 fl/,9;-'[*(*„ + dtJh, Yj(h, k))] |.=0 (r s* 2)

7=1

and observe that the right-hand side of this expression will only depend on the

derivatives d£Y¡(h,0) for p < r — 1. Consequently, in a step-by-step fashion we may

reduce the problem to the case r = 0, which is already known from (13). (The

reduction from r = 1 to r — 0 has actually been performed above.) The structure of

this reduction process is closely related to the set of Volterra-trees TV. In fact, a

tedious induction argument (omitted here), which is based on Faà di Bruno's

formula [1, p. 823] and similar to the proof of Theorem 6 in [8], shows

-Ta;y,(/z,o)= 2 qp/(0*(0*Bn(O (r>i),r<ETV

int(f) = r

where again tp, is given by Definition 5 and $ only depends on the integral equation

(1).

So we have finally found a factorization of Yj(h,ic) and yx(h, k) into their

"Runge-Kutta parts" <p, and cp and the "integral equation parts" $:

(14) Yi(h,K)=y(xQ + cih)+ 2 Vii*)*^)***0***0.terv

yx(h,K) =y(x0 + h) + 2 ç>(/)»(/)Afi"(')ici",('>.1<ETV

Now (10) and (12) complete the proof. □

Remarks, (a) For a — 0 condition (10) is equivalent to the order conditions given

in [3].

(b) The number of order conditions which have to be satisfied to obtain a

prescribed local order strongly depends on the exponent a and tends to infinity as

a -» -1. (Trees grow into the sky.)

This indicates that the construction of (noncollocation) Runge-Kutta methods

becomes increasingly complicated for negative values of a. On the other hand, it will

be comparatively easy to construct high-order explicit methods for positive a (see

Section 4).

(c) Figure 4 illustrates how the number e of Theorem 6 depends on a. Consider the

straight line L: fin + (1 + a)int = p. Then e is the smallest vertical distance between

L and the points with integer coordinates above L. (This follows from (10) and (14).)

We have 0 < e *£ min(l, 1 + a}.

Special values for e are:

e = 1 for integer a,

e = l for« = -i, {, I,....


94 CH. LUBICH

P LÎ+5"

FlGURE 4

The second part of this section is devoted to the study of the local error for the

first steps near 0, where the exact solution is usually not smooth. However, the

representation of the solution near 0 given by Theorem 2 or Corollary 3 permits

essentially the same derivation of the order conditions as before. We shall give these

order conditions for the sake of completeness even if their practical value seems a

little doubtful.

In this case the order conditions and the coefficients of the method will depend on

n, the step-number.

We fix n s* 0 and rewrite (1) as

(15) y(x) = F„(x)+ f\x-s)aK(x,y(s))ds for x G [nh, x],Jnh

where Fn(x) =f(x) + /0"*(x - s)aK(x, y(s)) ds.

Applying one step of the Runge-Kutta method (3) to ( 15), we obtain

m

(16) Yj = Fn((n + c,)h) + hx+° 2 «,-,*((« + dtJ)h, Yi),7=1

m

Ä+i = Fn((n + 1)A) + hx+« 2 b,K((n + e,)h, Y,).i=i

Now the following set of trees will be of importance.

Definition 7. Let TV0 denote the set of all trees which may or may not have an

index x or a attached to any of their final nodes. Trivially we have TV C TV0.

The definitions of fin(z) and int(i) remain the same as in Definition 4 with the

difference that we agree upon counting a-nodes as interior nodes.



^J ■VT»V

Figure 5

For the tree of Figure 6 we have fin(/) = 4, int(z)

Figure 6

The following definition is an extension of Definition 5.

Definition 8. Let

Jn(!x,l2)=f(l-s)as<in + sf+a)hdS,

y¡ = yy> = (n + c,)l+a, y = y») = (n+\)x+y

We define functions <p. = y\n\ R (/ = 1,... ,m) recursively by

m

<p,(k*,T'',T„'2]) = 2 a^/j^-jMuM^^'yh (k,ix,i2>oy7=1

9(k*,T\T¿])= 2 VM'y/2-•/„(/,,/2)y\i=i

and

9,(0= 2VÍ«ÍWi)-?y(',).7=1

<p(0= ÏVW'M)-9i('()i=i

for t = [Tk, T\ l£, f„. . .,g (?, # T„ T, T„, <7 > 1).

Remark. The restriction of «p'"', <p(n) to TF (l2 = 0) yields the functions <p¡, <p of

Definition 5.

The order conditions near 0 are now given by

Theorem 9. Consider an integral equation (1) such that the solution y(x) has an

asymptotic expansion in mixed powers ofx and x' +° as x -» 0 (cf. Corollary 3). Then

qf"\t) = 0 for all t G TV0with fin(z) + (1 + o)int(i) 0 which depends on a.

Compared to Theorem 6 this result states that the lack of smoothness near 0 can

be compensated by satisfying certain additional order conditions.

The proof is similar to that of Theorem 6. Instead of the solution v(x) one uses

the smooth function Y(zx, z2) with y/x) = F(x, xx+a) of Theorem 2 (or Corollary

3). We omit the details.

4. Examples. In this section we will use the order conditions to derive various

examples of Runge-Kutta methods (3).

A Runge-Kutta method (3) (resp. (8)) with (9) will be said to have local order p if

its coefficients satisfy condition (10).

According to formula (8) the internal stages Y, (i — I,_m) can be interpreted

as approximations to y(x0 + c,h), and yx approximates y(x0 + h). So it appears

natural to choose

(17) d:j = c„ e,= l (i,j=i,...,m).

Runge-Kutta methods whose coefficients satisfy (17) are called Pouzet-type methods

[3], [12]. The following theorem is an extension of Theorem 3.1 in [3]. Here T C TV

denotes the set of the Volterra-trees without x-nodes.

Theorem 10. Let a¡: and b¡ (i, j = I,...,m) represent a Pouzet-type method (9),

(17).//

(18) <p(t) = 0 forallt G Twithim(t) + (1 + a)int(f) *£/>

(where tp(z) is given by Definition 5), then the method has local order p.

Proof. The proof is analogous to the proof of Theorem 3.1 of [3].

(t.,t. e TV)

* ?■

tr

t, • .*r

Figure 7



Under the condition (17) one can see from Definition 5 that many trees in TV

have identical <p. Such pairs are sketched in Figure 7.

Hence, for any tree t G TV we can construct a tree t' G T such that qp(z) = <p(t'),

int(z) = int(z'), fin(z) s* fin(z'). Therefore (18) implies (10). D

Collocation Methods for x0 > 0. We now consider a class of implicit Pouzet-type

methods which satisfy the order conditions (10) in a trivial way. Choose distinct c,

(i — 1_,m) and determine the coefficients a.,., b, (i, j = 1,... ,m) from the

Vandermonde-type conditions (cf. [6, p. 142])

<P,(W\) =0, «p([t']) = 0 (/= l,...,m;/ = 0,...,w- 1),

i.e. (see Definition 5)m m

(19) 2 a,^ = J(iyi+x+a, 2 b,c\ = J(i) (1 = 0.m- 1).7=1 i=l

This means that each of the product quadrature formulae in (3) is exact for

polynomials of degree < m.

Definition 5 shows that the corresponding Pouzet-type method satisfies (18) with

p — m and hence, by Theorem 10, has local order m. But we have even

(20) <p,(?) = 0, <p(/) = 0 for all trees Z G TKwithfin(z) =£ m - 1,

and by (12) and (14) this implies (in the notation of Section 3)

(21) Y,-y(x0 + c,h) = O(hm+x+a) (t=l,...,m),

ft -y(xQ + h) = 0(hm+x+a).

Remark. For ordinary differential equations (i.e. the case where a = 0 and

K(x, s, y) does not depend on x) condition (19) is equivalent to stating that the

Runge-Kutta method is a collocation method (cf. [10]). For arbitrary a, if cm = 1,

Pouzet-type methods satisfying (19) can be interpreted as collocation methods in the

sense of [3, Section 4] and [4].

There is even local superconvergence en miniature:

Proposition 11. Let c,,...,cm be distinct nodes such that the error of the corre-

sponding product quadrature formula is of order

m

(22) hx+" 2 b,g(c,h) -f(h- s)ag(s) ds = 0(h«+x+«)

for smooth g(x), where q > m + 1.

Then the local error of the corresponding Pouzet-type method, whose coefficients are

determined by (19) and (17), satisfies

m + 2(1 + a) for a < 0,y\ ~y{x0 + h) = 0(hr), where r

1 m + 2 + a for a > 0.

Proof. The assumption on the product quadrature rule implies

m

<p({r'})= 2 b,c', - /(/) = 0 for/ = 0,...,<7-l.i=i

As in the proof of Theorem 10, this condition yields ip(t) = 0 for all t G TV with

fin(0 < q — 1, int(r) = 1. Together with (20), this gives the result via (12) and (14).


98 CH. LUBICH

As an illustration consider Figure 8 where the (fin, int)-coordinates of the trees with

nonvanishing O

• • •

• • •

O ï 2 '-'-*lnt o'-1-5-V~-Int

Figure 8

Remark. For a = 0 the local error is actually 0(hq+x). (For ordinary differential

equations this is proved e.g. in [10]; see also [6, p. 143]. By Theorem 10 the error of

the corresponding Pouzet-type method is of the same magnitude.) An analogous

statement does not hold for arbitrary a. As the following example demonstrates, the

result of Proposition 11 can in general not be improved for negative a. (It is obvious

from Figure 8 how a stronger result can be obtained for a > 0 and q > m + 2.)

Example. Let o = -{-, m = 2. Choose cx,c2 as the zeros of the polynomial

x2 — 8/7 ■ x + 24/105 (Gauss nodes), and determine b,, b2 from (19). Then we have

q — A. However, the local error of the corresponding Runge-Kutta method (19), (17)

is only 0(h3), because the order condition for the tree [[t2]] is not satisfied. D

Collocation Methods Near 0. For the approximation of the nonsmooth solution

near 0 the same concept as above leads to nonpolynomial collocation methods. For

a = -j, such methods have recently been put forward in [13], [5].

Choose distinct c, (i = 1.m), and determine the coefficients a¡¡, b, (i, j =

1,... ,m) from the conditions

<Pzn)([A^J) = o,

<P<">([t\ t'J\) =0 (i = \,...,m; /, + (1 + a)l2<p),

i.e. (see Definition 8)

m

2^J1(« + c,)/2(1+a)=/n(/1)/2)c,'.+,+a(« + c(),j(1+a) (i = l,...,m),

7=1

m

2 bjcHn + c,)w+a) =Jn(lx, l2) for /, + (1 + o)/2 </>,i-i

where m and p are related in such a way that the system of linear equations has a

unique solution.

If djj, e¡ are chosen according to (17), a similar argument as in the smooth case,

which is now based on Theorem 9, yields (the notation is as in formula (16))

m+2(1+a» i»-i4

fin"

H2+a -\

m-1-



Yj-y((n + c,)h) = 0(h"+e) (i = l,...,m),

yn+x-y((n+l)h) = 0(h^)

for some e > 0.

Explicit Runge-Kutta Methods for x0 > 0. To begin with, the explicit Euler method

reads

yn+\ =Fn(xn+x) + y^-a-hx+aK(xn+x,xn,yn).

The method has local order 1. It satisfies (19), and (21) shows that the local error is0(h2+a).

Since the number of order conditions (10) depends strongly on a, there is no point

in constructing high-order explicit methods which have the same local order for all a

(as it was for Euler's method above). It is more promising to construct methods for

special, practically important values of a.

For a = 0, various examples are given in [3]. For a = - { we begin with a negative

result.

Proposition 12. There is no 2- (resp. 3-, 4-) stage explicit Runge-Kutta method (3),

(9) for a = -4_ having local order p = 2 (resp. f ,3) (i.e. local error 0(hp+x/2)).

■ i < v S </ Y í v(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)

Figure 9

Proof. An explicit method of local order 2 has to satisfy at least the following

order conditions (see (18) and Figure 9):

m

(0 2 b, = 2,i=im 4

(Ü) 2 bjC, = -,1 = 2

("i) 2b,\2a,JCj-&/2)=0.i=2 \y=2 /

For m = 2, (iii) yields b2c\/2 = 0 which contradicts (ii). For an explicit method of

order § the following order conditions also have to be satisfied:

(iv) 2 b,c2 = —,1 = 2

m i-l / 7-1 a \

(v) 2 2 bjOjA 2 a]kck - -c)/2 = 0.¡=3;=2 \k=2 I

If m = 3, (v) implies b3a32c2/2 = 0. Then (iii) reads 62C2/2 + b3cj/2 = 0. Inserting

this relation in (ii) and (iv), we obtain

V3c2-1/2(4/2 - cY2) = |, - b3cV2(cx/2 - cY2) = i|,

which is a contradiction.


100 CH. LUBICH

Finally, for/? = 3 also the following order conditions have to be satisfied:

(vi)

(vu)

(viii)

m I /'— 1

2 Vz 2 a,jCji=2 \./ = 2

m I i — 1

2 *,■ 2 a/7c7 -1=2 \/=2

m i— 1 >—I

3

16

Ï5'

,3/2

,5/2

0.

A-l

2 2 2 V/yöy* 2V/-^í/!1 = 4 7 = 3 A = 2 \ /=2

If m = 4, (viii) implies b^aAiaJ2c\/2 = 0, and the contradiction follows in a similar

but more technical way as above. D

If we choose m = 3, c2 = §, c3 = 1, />2 = 0, then (i), (ii), (iii) and Theorem 10 yield

Example 13. The following coefficients represent a 3-stage explicit Pouzet-type

method for a = - { of local order 2 (i.e. local error 0(h5/2)).

0

2/3

1

0

2/6/3 0

0 2

2/3 0 4/3 b,

In the notation of (3) the method reads

Y^Fn(x„),

Y2 = F„(x„ + 2/1/3) + 2v'6/3-/z'/2 • /V(x„ + 2A/3, *,„ K,),

F3 = F„(x„ + A) + 2hx/2-K(x„ + h, x„ + 2A/3, F2),

Ä+i = £(*„ + A) + 2/3 -/z'/2 • /V(x„ + A, x,„ F,)

+ 4/3 • A1/2 • yV(x„ + A, x„ + A, F3).


method for a = - | of local order 3 (i.e. local error 0(h1/2)).

0

1/2

1/2

1

1

3

-4\/2"45

0

2/2

'43

157T 0

48 + 8/2

45

_1615

0

36-4^/2"^5

4

5



where

'53'42

'43

'41

(7'22

54

8 _32

3 + 27a

1.84296978...,

zz42 = 6.267309622...,54

a42- 43-2.424339842....

The method was derived in the following way: Choose c, = 0, c2 = c3 — Y c4 = c5

= 1, b2 = 0, b4 = 0, a52 - 0. Then (i), (ii) and (iv) give bx,b3 and b5. (iii) and (vi)

imply a32 = 2{2 /3, whence a53 and a54 can be obtained from (iii) and (vii). Now the

value for a42 follows from (viii), and (v) gives a43. Finally, the coefficients a2x,... ,a5]

are obtained from (9). By Theorem 10 the method has the asserted local order.

Proposition 12 and the foregoing examples indicate that the construction of high

order explicit Runge-Kutta methods for negative exponents a is by far more

complicated than for a = 0. The converse situation holds for positive values of a.


method for a = 1 of local order 4 (i.e. local error 0(hs)).

0

1/2

0

1/8 0

1/6 1/3 b,

In the notation of (3) the method reads

Yx = F„(x„),

Y2 = Fr[xn + \)+~K[xn + \,xn,Y,],

yr,+ ̂ = F„(x„ + h)+ ^K(x„ + h, x„, Yx) + y*(x„ + h, xn + |, Y2).

It was derived from (9) and the following order conditions (see Figure 9)

(i)A, + A2 = |,(ii)b2c2 = i,

(iv) b2c\ = y2.

By Theorem 10 the method has the asserted local order.


method for a = 1 of local order 5 (i.e. local error 0(h6)).

0

2/51

0

2/25

-1/4

0

3/4

1/8 25/72 1/36 b,


102 CH LUBICH

The method was derived from (9) and the following order conditions after choosing

c3 = 1 (see Figure 9)

(i)bx + b2 + b3 = y

(ii)b2c2 + b3c3 = i,

(iv)b2c2 + b3c2 = -rj,

(ix)Z>2C2 + Vl = 2t),(m)b2(-ici) + b3(a32c2-ic¡) = 0.

Acknowledgement. The author would like to thank E. Hairer, P. Kaps and G.

Wanner for helpful discussions.

Universität Heidelberg

Institut für Angewandte Mathematik

Im Neuenheimer Feld 293

6900 Heidelberg I, Federal Republie of Germany

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